Jacobian versus Infrastructure in Real Hyperelliptic Curves

Speaker:

Monir Rad

Date:

Thu, Nov 17, 2016 - Sat, Dec 17, 2016

Location:

PIMS, University of Calgary

Conference:

PIMS CRG in Explicit Methods for Abelian Varieties

Abstract:

Hyperelliptic curves of low genus are good candidates for curve-based cryptography. Hyperelliptic curves comes in two models: imaginary and real. The existence of two points at inﬁnity in real models makes them more complicated than their imaginary counterparts. However, real models are more general than the other model, every imaginary hyperelliptic curve can be transformed into a real curve over the same base ﬁeld Fq , while the reverse process requires a larger base ﬁeld.

Real hyperelliptic curves have not received as much attention by the cryptographic community as imaginary models, but more recent research has shown them to be suitable for cryptography. Real models admit two structures, the Jacobian (a ﬁnite abelian group) and the infrastructure (almost group just fails associativity). In this talk, we explain these two structures and compare their arithmetic based on some recent research. We show that the Jacobian makes a better performance in the real model. We also conﬁrm our claim with some numerical evidence for genus 2 and 3 hyperelliptic curves.